3.788 \(\int \sqrt{\sqrt [4]{x}+x} \, dx\)

Optimal. Leaf size=59 \[ \frac{2}{3} \sqrt{x+\sqrt [4]{x}} x+\frac{1}{3} \sqrt{x+\sqrt [4]{x}} \sqrt [4]{x}-\frac{1}{3} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt [4]{x}}}\right ) \]

[Out]

(x^(1/4)*Sqrt[x^(1/4) + x])/3 + (2*x*Sqrt[x^(1/4) + x])/3 - ArcTanh[Sqrt[x]/Sqrt
[x^(1/4) + x]]/3

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Rubi [A]  time = 0.113001, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{2}{3} \sqrt{x+\sqrt [4]{x}} x+\frac{1}{3} \sqrt{x+\sqrt [4]{x}} \sqrt [4]{x}-\frac{1}{3} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt [4]{x}}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x^(1/4) + x],x]

[Out]

(x^(1/4)*Sqrt[x^(1/4) + x])/3 + (2*x*Sqrt[x^(1/4) + x])/3 - ArcTanh[Sqrt[x]/Sqrt
[x^(1/4) + x]]/3

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Rubi in Sympy [A]  time = 6.50314, size = 49, normalized size = 0.83 \[ \frac{\sqrt [4]{x} \sqrt{\sqrt [4]{x} + x}}{3} + \frac{2 x \sqrt{\sqrt [4]{x} + x}}{3} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{x}}{\sqrt{\sqrt [4]{x} + x}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**(1/4)+x)**(1/2),x)

[Out]

x**(1/4)*sqrt(x**(1/4) + x)/3 + 2*x*sqrt(x**(1/4) + x)/3 - atanh(sqrt(x)/sqrt(x*
*(1/4) + x))/3

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Mathematica [A]  time = 0.0404164, size = 57, normalized size = 0.97 \[ \frac{3 x^{5/4}-\sqrt{x^{3/4}+1} \sqrt [8]{x} \sinh ^{-1}\left (x^{3/8}\right )+2 x^2+\sqrt{x}}{3 \sqrt{x+\sqrt [4]{x}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x^(1/4) + x],x]

[Out]

(Sqrt[x] + 3*x^(5/4) + 2*x^2 - Sqrt[1 + x^(3/4)]*x^(1/8)*ArcSinh[x^(3/8)])/(3*Sq
rt[x^(1/4) + x])

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Maple [C]  time = 0.108, size = 342, normalized size = 5.8 \[{\frac{2\,x}{3}\sqrt{\sqrt [4]{x}+x}}+{\frac{1}{3}\sqrt [4]{x}\sqrt{\sqrt [4]{x}+x}}+{\frac{-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt [4]{x} \left ( 1+\sqrt [4]{x} \right ) ^{-1}}} \left ( 1+\sqrt [4]{x} \right ) ^{2}\sqrt{-{\frac{1}{{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( \sqrt [4]{x}-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( 1+\sqrt [4]{x} \right ) ^{-1}}}\sqrt{-{\frac{1}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( \sqrt [4]{x}-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \left ( 1+\sqrt [4]{x} \right ) ^{-1}}} \left ( -{\it EllipticF} \left ( \sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt [4]{x} \left ( 1+\sqrt [4]{x} \right ) ^{-1}}},\sqrt{{\frac{ \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }{ \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}} \right ) +{\it EllipticPi} \left ( \sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt [4]{x} \left ( 1+\sqrt [4]{x} \right ) ^{-1}}},{\frac{{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{ \left ( -{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }{ \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}} \right ) \right ){\frac{1}{\sqrt{\sqrt [4]{x} \left ( 1+\sqrt [4]{x} \right ) \left ( \sqrt [4]{x}-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \left ( \sqrt [4]{x}-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^(1/4)+x)^(1/2),x)

[Out]

2/3*x*(x^(1/4)+x)^(1/2)+1/3*x^(1/4)*(x^(1/4)+x)^(1/2)+(-1/2-1/2*I*3^(1/2))*((3/2
+1/2*I*3^(1/2))*x^(1/4)/(1/2+1/2*I*3^(1/2))/(1+x^(1/4)))^(1/2)*(1+x^(1/4))^2*(-(
x^(1/4)-1/2+1/2*I*3^(1/2))/(1/2-1/2*I*3^(1/2))/(1+x^(1/4)))^(1/2)*(-(x^(1/4)-1/2
-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(1+x^(1/4)))^(1/2)/(3/2+1/2*I*3^(1/2))/(x^(1
/4)*(1+x^(1/4))*(x^(1/4)-1/2+1/2*I*3^(1/2))*(x^(1/4)-1/2-1/2*I*3^(1/2)))^(1/2)*(
-EllipticF(((3/2+1/2*I*3^(1/2))*x^(1/4)/(1/2+1/2*I*3^(1/2))/(1+x^(1/4)))^(1/2),(
(-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/
2)))^(1/2))+EllipticPi(((3/2+1/2*I*3^(1/2))*x^(1/4)/(1/2+1/2*I*3^(1/2))/(1+x^(1/
4)))^(1/2),(1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)),((-3/2+1/2*I*3^(1/2))*(-1/2-1
/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x + x^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + x^(1/4)),x, algorithm="maxima")

[Out]

integrate(sqrt(x + x^(1/4)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + x^(1/4)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\sqrt [4]{x} + x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**(1/4)+x)**(1/2),x)

[Out]

Integral(sqrt(x**(1/4) + x), x)

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GIAC/XCAS [A]  time = 1.00642, size = 61, normalized size = 1.03 \[ \frac{1}{3} \, \sqrt{x + x^{\frac{1}{4}}} x^{\frac{1}{4}}{\left (2 \, x^{\frac{3}{4}} + 1\right )} - \frac{1}{6} \,{\rm ln}\left (\sqrt{\frac{1}{x^{\frac{3}{4}}} + 1} + 1\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | \sqrt{\frac{1}{x^{\frac{3}{4}}} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + x^(1/4)),x, algorithm="giac")

[Out]

1/3*sqrt(x + x^(1/4))*x^(1/4)*(2*x^(3/4) + 1) - 1/6*ln(sqrt(1/x^(3/4) + 1) + 1)
+ 1/6*ln(abs(sqrt(1/x^(3/4) + 1) - 1))